## Saturday, April 26, 2014

### Similar Triangles, 2

Category: Plane Geometry

"Published in Vacaville, California, USA"

A street light is 15 ft. directly above the curb. A man 6 ft. tall starts 10 ft. down street from the light and walks directly across the street which is 10 ft. wide. When he reaches the opposite curb, what is the distance between the initial and final positions of the tip of his shadow?

Solution:

To illustrate the problem, it is better to draw the figure as follows

 Photo by Math Principles in Everyday Life

Let x = be the length of his shadow at the 1st curb.

By using similar triangles

If a man walks 10 ft. directly across the street which is at the opposite curb, the figure above becomes

 Photo by Math Principles in Everyday Life

In order to solve for the distance between the initial and final positions of the tip of his shadow, let's label further the above figure as follows

 Photo by Math Principles in Everyday Life

Let a = be the distance of a man to the street light from 2nd curb
b = be the length of his shadow at the 2nd curb
y = be the distance between the initial and final positions of the tip of his shadow

By Pythagorean Theorem

By using similar triangles

Since the 1st Curb is parallel to the 2nd Curb, then the small and large triangles are similar to each other. If the small triangle is an isosceles right triangle, then the large triangle is also an isosceles right triangle.

By Pythagorean Theorem

which is the same as the length of a man's shadow at the 1st Curb.

Therefore, the distance between the initial and final positions of the tip of his shadow is